Fractions as Operators (Dot Arrays)

Here’s a collection of strings written by teacher participants at the Summer Institute at Math in the City (City College, NY).

When students share their strategies, you might ask, “How do you know?  How are you seeing it on the array?”  Then circle or shade what they saw.  Remember to open it up to other ways of seeing, “Did anyone think of it differently? Oh great. Ronald, what did you see?”  Then the second student’s strategy or envisioning is shown on a different array.  I like to print several copies of the array and have them ready to go up.  Otherwise, it takes too long to draw the dots each time. Continue reading “Fractions as Operators (Dot Arrays)”

Multiplying fractions: Why context matters

Our fifth grade team was trying to encourage students to use a visual model to represent their thinking when they multiplied fractions. So many students were so fast — multiplying the numerators, then multiplying the denominators — but had no context and demonstrated very little number sense. Did their answer, the product, make any sense? What would happen to the size of the first fraction as it was multiplied by the second fraction? Was the product bigger or smaller than the fractions? Should it be? We saw students who were simply carrying out some steps without thinking about what multiplying fractions really means.

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Photo number strings for multiplication

Here are two photos I snapped as I walked by a 99 cent store in LA. Beautiful arrays, no?  Image

I am thinking about how to use these kinds of images as the anchors for number strings, particularly for intervention work with older students.  Sometimes older kids need work thinking about multiplication, but in an age-appropriate way.  What kind of questions do you think of with this image?  One could most simply begin by asking what kids noticed about the image.  That would bring most of the interesting mathematics forward, I think. Beginning perhaps with how many boxes of hot chocolate do you see (nice numbers)?  And then, considering this is a 99 cent store, how much would it cost to buy all of this chocolate.  It reminds me of some work that Pamela Harris suggests in her book on Powerful Numeracy, in which she asks kids what is 99 plus any number?  A 99 cent store is a great way to think about what is 99 times any number?

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A Dilemma with Models

A 5th grade teaching team I work with recently raised the issue of how to model the problem 1/2 x 3/10 on an array. They saw their students use a variety of models and one teacher got “bothered” by how some of the models felt “imprecise” or “not to scale.” We had a conversation together about this issue.  To prepare myself for the conversation I did a little sketching of possible models that kids might use.  What do you think?

Fraction Models_Multiplication